If \( \displaystyle \alpha = \int\limits_{\frac{1}{2}}^{2} \frac{\tan^{-1} x}{2x^2 - 3x + 2} \, dx\), then the value of \( \sqrt{7}\,\tan\!\left(\dfrac{2\alpha \sqrt{7}}{\pi}\right) \) is __________.
(Here, the inverse trigonometric function \( \tan^{-1} x \) assumes values in \( \left( -\dfrac{\pi}{2} , \dfrac{\pi}{2} \right) \).)
